Dyr og Data
Statistical thinking — exploratory data analysis
Gavin Simpson 
Aarhus University
2024-09-19
Barcharts
Bar charts are a simple way of showing & comparing means or other values between groups of data
Top of the box is drawn at the data values you wish to plot, bottom at 0
Dotplots
However, barcharts are wasteful & often don’t focus on the real information you want to convey
Dotplots show the same information but focus on the difference in the variable plotted
Boxplots
A boxplot is a useful summary of samples with \(n > 8\) observations
Shows the median, the upper and lower hinges & whiskers, plus any observations that lie beyond the whiskers
- Median is robust to outliers
- Hinges are ~ lower & upper quartiles
- Whiskers extend to at most \(\pm 1.58 \times IQR / \sqrt{n}\)
- Points beyond whiskers are shown individually; not outliers
Boxplots
Useful to Compare observations from 2 or more groups
- Width of box is proportional to \(n\)
- Notches around median are ~ 95% confidence intervals for difference of medians
- If notches overlap, fail to reject null hypothesis that observations drawn from populations with same median
Histograms
Histograms; a graphical representation of the frequency or density distribution of data
- Data are assigned to determined bins
- Values of a discrete variable usually determine the bins
- Number of bins is determined by the bin width
- Area under a histogram sums to 1 for histogram showing densities
- Often, histograms drawn as frequency histograms; area no longer sums to 1
Histograms
Choosing the bin width is a critical step in drawing a histogram1
- \(k = \sqrt{n}\)
- Sturge’s: \(k = \frac{\mathrm{range}(x)}{(\log_2n + 1)}\)
- Scott’s: \(k = \frac{3.49\hat{\sigma}}{n^{1/3}}\)
- Freedman-Diaconis: \(k = 2 \frac{\mathrm{IQR}(x)}{n^{1/3}}\)
\(\hat{\sigma}\) is the estimated SD, \(n\) number of observations, & \(\mathrm{IQR}\) the interquartile range
Quantile-Quantile plots
Quantile-quantile (QQ) plots are useful to determine if a sample is normally distributed
Draws quantiles of the data & a reference distribution. If normally distributed, points should fall on line through upper & lower quartiles of both distributions
100 random draws from a \(t_3\) distribution — heavy tails compared to normal
Scatterplots
Thus far dealt only with univariate data displays. Other displays needed for bivariate and multivariate data
A scatterplot displays the relationship between two variables, \(x\) and \(y\) say
Each point on the plot represents the value of \(x_i\) and \(y_i\) for a single observation \(i\)
Simpson’s paradox
Important to plot data so you aren’t surprised when you model it
Violin plots
Violin plots can be thought of as a combination of a boxplot and a density plot
Raincloud plots
Boxplots and violin plots can be criticised because they don’t show the data
Raincloud plots are an alternative that does — the dots are binned like a histogram
Dotplots
Dotplots are a related graph that shows the data
Beeswarm plots
Beeswarm plots show all the data in a compact deisplay and avoid points overlapping
